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Hey, PF
I'm reading the following derivation of least squares, and I'm trying to figure out how the author went from the last step at the bottom of pg. 7 to the final equation (11) at the top of pg. 8.
[http://isites.harvard.edu/fs/docs/icb.topic515975.files/OLSDerivation.pdf]
More specifically, why is the denominator a difference of two terms? Aren't the terms in the denominator summed in the prior step?
I would expect the answer to be
$$
b_1=\dfrac{\displaystyle \sum_{\textrm{i=1}}^{n}y_ix_{i}-\left(\frac{1}{n}\right)\left(\sum_{\textrm{i=1}}^{n}y_i\sum_{\textrm{i=1}}^{n}x_{i}\right)}{\displaystyle\sum_{\textrm{i=1}}^{n}x_{i}^2+\left(\frac{1}{n}\right)\left(\sum_{\textrm{i=1}}^{n}x_{i}\right)^{2}}
$$
Note: I'm no statistician, but I thought you guys might be more familiar with this derivation.
I'm reading the following derivation of least squares, and I'm trying to figure out how the author went from the last step at the bottom of pg. 7 to the final equation (11) at the top of pg. 8.
[http://isites.harvard.edu/fs/docs/icb.topic515975.files/OLSDerivation.pdf]
More specifically, why is the denominator a difference of two terms? Aren't the terms in the denominator summed in the prior step?
I would expect the answer to be
$$
b_1=\dfrac{\displaystyle \sum_{\textrm{i=1}}^{n}y_ix_{i}-\left(\frac{1}{n}\right)\left(\sum_{\textrm{i=1}}^{n}y_i\sum_{\textrm{i=1}}^{n}x_{i}\right)}{\displaystyle\sum_{\textrm{i=1}}^{n}x_{i}^2+\left(\frac{1}{n}\right)\left(\sum_{\textrm{i=1}}^{n}x_{i}\right)^{2}}
$$
Note: I'm no statistician, but I thought you guys might be more familiar with this derivation.